Closeness of Classical Orbits and Factorization of Radial Schroinger Equation

Key words: 
• Bertrand ′s theorem /
• closed orbits /
• creation and annihilation operator

Abstract: It is shown that for a particle with suitable angular momenta in the screened Coulomb potential or isotropic harmonic potential, there still exists closed orbits rather than ellipse, characterized by the conserved perihelion and aphelion vectors, i.e., extended Runge Lenz vector, which implies a higher dynamical symmetry than the geometrical symmetry SO 3. For the potential, factorization of the radial Schrdinger equation to produce raising and lowering operators is also pointed out.